Really stupid suggestion but couild it be that the transistor has all three pins along the line of a continuous circuit on the breadboard. That would make the whole transistor look shorted and make the led light.

Don't think so. The breadboard layout looks correct even going by a badly focused photo.

 

I did wonder which way the connections ran on the breadboard.

Perhaps mrSilkie would provide a better (plan) photo?

 

The title has been poorly constructed and my circuit probably has been aswell.

I got an arduino earlier this week and it has been great, it is the ultimate platform for learning digital electronics, it doesn't just drop me off in the deep end and i already have some coding experience so the learning curve isn't too high.
Anyways, tonight i learnt about relays and transistors, turns out i have a transistor in my little arduino starter kit so i made a circuit with 5v connected to the collector, and a resistor and LED on the emitter end, i'm just using the 3v wire on my arduino to apply a current to the base. My problem is that even when the base isn't getting a current, the led is still dimly lit. Is this because the transistor has a maximum voltage? other then that, i have no clue what could be causing it.

http://www.technologystudent.com/elec1/transis1.htm
my transistor is similar to the bottom transistor in the animated gifs seen at the bottom of the screen.

Also, because I've only been learning about electronics for a week or so, are there any good books that will show me the absolute basics?

Because you mentioned relays in your text - I'm wondering if you have at any time driven a relay without a back emf catch diode, having done so could have rendered your transistor leaky or even short-circuit!

 

heres a couple of schematics just as examples using a pnp and npn transistor to switch a load-with a pnp if you connect a resistor as shown from the base to the collector that will ensure the transistor is switched off with no signal applied-leave it floating and and the transistor depending on the type/gain of the device may still switch on -same with an npn type-the pnp requires a negative going signal to switch it on while the npn requires a positive going signal for switching